Finding Limits Graphically

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Question

Finding Limits Graphically

Graph the functions in Exercises 9 and 10. Then answer these questions.

f(x) = {x,−1 ≤ x < 0, or 0 < x ≤ 1

1, x = 0

0, x < −1 or x > 1

d. At what points does the right-hand limit exist but not the left-hand limit?

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says sketchy graph of F of X, which is equal to -2 X, when -3 is less than or equal to X, which is less than -1, or when -1 is less than X, which is less than or equal to 1, and 0 when X is equal to -1, and 2, when X is less than -3, or when X is greater than 1. And we need to determine the points where the left-hand limit exists, but not the righthand limit. Below the problem we're given an empty graph on which to plot our function. So the first part of this problem, we need to sketch the graph of F of X. And so we're going to treat each piece individually. So, we're gonna start off with the first piece, and so we're gonna have F of X. E equal to minus 2 X. And so we need to evaluate this expression at the endpoints of the two intervals that we were given for the first piece. So, the first endpoint is going to be when X is equal to -3, so we'll calculate F of -3, and that's gonna be equal to -2, multiplied by -3, which is equal to 6. The next point is going to be when X is equal to -1, so we'll have F of -1, and that's going to be equal to -2, multiplied by -1, which is equal to 2. And the last point that we need to look at is when X is equal to 1, so F of 1 is going to be equal to -2, multiplied by 1, which is equal to -2. So, we're not going to plot these points on our graph. So the first point that we're gonna plot is going to be X equal to -3, and our Y value is going to be equal to 6. And we're gonna plot that with a solid dot. Since the value of X equal to m3 is included in our interval. Now when we look at the next point, which is going to occur when X is equal to -1 and Y is equal to 2, here we're going to need to use an open dot because the value of X equal to -1 is not included on either of the two intervals that we're looking at. So we're going to have an open dot here. That means our function is going to approach the value of X equal to -1, but never actually take on that value. So now I have these two points, plot it, I'll just now connect them with A straight line. So that's gonna be the graph of the first interval. Now, for the 2nd interval, we're again going to start. At X equal to -1, 2. And again, that's still going to be an open point, since X equal to -1 is not included in the interval. And that's gonna go to our second point, which is going to be at 1-2. And that is going to be a solid dot, since X equal to 1 is included in the interval. And so now we want to connect these two points. Together with a straight line as well. Now if we look at the next piece of our piecewise function, we see that we have F of X equal to 0 when X is equal to -1. So here we just have a single point to plot, and that is going to be located at the point of -1.0, and we'll plot that with a solid dot. And now the last piece is when F of X is equal to 2, when X is less than minus 3, and X is greater than 1. So here we're going to have a horizontal line at Y equal to 2. So, for the first interval, when X is less than -3, we're gonna have an open circle at X equal to -3, since X equal to -3 is not included in our interval, and then we'll have a horizontal line going to the left from that point. Then on the other side, for our other interval, we're gonna have an open dot at 8 X equal to 1. And Y equal to 2. And we're going to have a horizontal line going to the right from this point. And so the graph that we have on the screen here is the graph that the um question was asking for. So now we need to actually determine the points where the left hand limit exists, but not the right hand limit. And we're actually going to check the points where we actually have jumps on our graph. So, the first one we need to look at is when X is equal to -3. So, we're gonna look at the limit as X approaches -3 from the left of F of X. And if we approach X equal to minus 3 from the left, we see that our function F of X is approaching a value of 2. And now we need to look at the limit as X approaches minus 3 from the right of F of X. And here if we look at our graph, we see that Our function F of X is approaching a value of 6. That limit is going to be equal to 6. So, even though we have different values here, both our left hand limit and our right hand limit exist at this point. The next point we're going to want to look at is when X is equal to -1. So we'll look at the limit as X approaches -1 from the left of F of X. And if we approach 1 from the left and follow our curve, we see that we're approaching a value of 2, so this limit is going to be equal to 2. We also need to look at the limit as X approaches -1 from the right of F of X. And again, Approaching the value of X equals 2-minus 1 from the right, we also get a value of 2, so that limit is also equal to 2. And the last point we want to look at is when X is equal to 1. So we'll look at the limit as X approaches 1 from the left. Of F of X And this is going to be equal to, and if we come over here to our graph, and if we approach the value of X equal to 1 from the left, we're approaching a value of -2, so that limit is going to be equal to -2. And then finally, we're gonna be looking at the limit as X approaches 1 from the right of F of X. And if we approach X equal to 1 from the right, our function F of X is approaching a value of 2, so that means that this limit is going to be equal to 2. So, in all three cases, we were able to find values for both the left-hand limit and the right hand limit. So they both exist. They weren't always the same value, but they both existed. And so that means that the points where the left hand limit exists, but not the right hand limit existing, is actually none. So that is going to be the second half of the second half of this problem answered. So, I hope that this explanation has been useful, and I'll see you in the next video.

Finding Limits Graphically

Graph the functions in Exercises 9 and 10. Then answer these questions.

f(x) = {x,−1 ≤ x < 0, or 0 < x ≤ 1
1, x = 0
0, x < −1 or x > 1

d. At what points does the right-hand limit exist but not the left-hand limit?

Key Concepts

Limits

Piecewise Functions

One-Sided Limits

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